Good question. I studied mathematics at Bristol and matriculated at Cambridge in 1990 to study Part III of their mathematical tripos. My favourite subjects were Algegraic Topology, Category and Topos Theory. I also enjoyed studying the foundations of mathematics, such as those of Frege, Russell, and Dedekind.

Pure mathematics in itself is not so useful, though I would say coding along the lines of functors is a good application. What other people closely associated with pure - applied, stats, physics, chem eng etc is that we studied someone that will to their dying breath unobtainable, whereas I can pick up a book on String Theory and get the gist. Because our brains are wired that way and Pure is, well, a pure subject requiring dedication to excel.

Two inter-connected things a pure mathematician need to avoid to be successful outside of academia are:-

1. Dont try to be the smartest person in the room. You did Pure so everyone will assume you are academically bright. In short, build a network of people you can work with and they can work with you.
2. Dont think tasks are either below you intellectually or that those who do those tasks are intellectually below you? Acknowledge the fact that any serious task needs the collaboration of a wide suite of skills of which you only have a fraction.

Finally, maintain the mystique of being other-worldly by not being the first to talk. Listen, consider and use your words wisely.

Hope that helps.

There was an interesting experiment in the 19th Century, between France and Germany.

The Germans focused on pure mathematics, that is Mathematics for its own sake, without being driven by the need for it to have any useful applications.

The French focused on applied Mathematics, that is Mathematics designed to solve practical problems.

German Mathematics soon outstripped the French, with Gottingen becoming the centre of a golden age of Mathematicians.

The French solved a lot of the problems of the time, mostly in probability, but the Germans solved problems that didn't exist, but were going to be vital as the 20th Century started.

Notably the Mathematics behind relativity had been worked out, as had general solutions involving abstract algebra.
New technologies such as radio and electricity required Mathematics that wasn't based on things you could see.
Numbers themselves turned out to be a whole lot more complicated and Maths moved into non-numerical fields (Boolean algebra and Graph theory)

Pure Maths at its heart is a quest for truth, it has an almost mystical element. Pure Maths is the equivilent of a mystery tour, you could end up anywhere, but the point of a journey is not the destination.

If you want a job where you can continue to do pure Mathemstics, the only real choice is to go into academia.

If you are willing to do other things, the fact is that you are prepared to do anything.

My BS was in pure Math. Then I did a few years as a graduate student doing pure math and philosophy without getting a degree. I spent several years programming after that, then several years in research on Wall Street, then (after going back to school) now teach at a university.

The fact is that a lot of companies will take a chance on you. When I was younger, their attitude was “He is young, cheap, and smart. He can learn what he needs here. If it doesn’t work out, we can fire him,”

I suspect many engineers are motivated by seeing their work turned into products. I base this on the complaints of colleagues that only worked on product proposals and never saw their ideas actually built. Pure mathematics is often perceived as more academic than practical, which makes it more difficult to explain than work that results in a tangible product. The key factor is likely what truly interests and motivates you. Pure science and academics can be very motivating for discovery of the unknown or exploring novel possibilities and concepts, while product design can be reduced to a series of similar incremental improvements, so motivation may be more based on the core of the career than the application and that may be easier to understand early in an engineering career than in pure mathematics.

The question would be more accurate to ask “Is engineering a wise career choice for the future?”

Engineering is hard enough. Super tough if you aren’t suited for the field you attempt to engage.

When I was going through school, there wasn’t a single EE student that could imagine making it as a mechanical engineer. Conversely, there wasn’t a single ME student that could imagine being a EE. We ragged on each other all the time (heat and beat engineer vs spark chaser, etc.), but in the end, either degree (and there is an array of engineering degrees) put you ahead in the general job market.

The answer to my restated question is “yes”. If you can gut it out. Most people that try fail in the first year. And it gets harder each successive year.

The terms pure mathematics and applied mathematics seem to have been invented in the 19th century, but they really aren't two different kinds of mathematics. The distinction is more of intent than of content. Pure mathematics isn’t intended to be applied to anything, but it might be sometime. Applied mathematics contains pure mathematics but it is mathematics that is used for something.

The applications are to some kind of science. Mathematics that is studied with the intent to be applied to natural science, biological science, or social science can be called applied mathematics. Scientists use a lot of mathematics in their work. If a scientist develops some new mathematics in the course of studying science, that could be called applied mathematics. If that same scientist goes a little further studying that mathematics without the intent of applying it, you could call that pure mathematics.

Pure mathematics is simply mathematics considered without regard to applications outside of mathematics. It may or may not have been applied. It may or may not be applied sometime in the future.

good question. Electrical engineering encompasses so many different areas: power, telecomm, computers, software, IOT, etc and there will always be a need for people who understand the concepts and how to manipulate them to get the desired results. at the same time, many areas of electrical engineered products are becoming commodity items such as andino and Build projects that you can do without understanding the nity gritty. Time to market is everything and these items help you get to market quickly.

my feeling is that if you get a good, broad EE degree, you can go in many directions for the foreseeable future and be able to follow the changes in the world that are sure to come

best of luck

Larry

Both are interesting ! But the correct question will be “which is more interesting for me/you?”. Let me explain (I'm by no means an expert) a distinction between pure and applied mathematics.

Pure Mathematics :- It's studying mathematics for its own sake. Pure mathematicians are just like painters and poets, their work may not have immediate impact on our society but their works are priceless, which may impact our society in future severely. But the language of their poetry is rock solid logic and their painting is abstract spaces. What is abstract spaces ? Just to give you an outline. Suppose you are standing on a particular football ground say X. You observed this field X and concluded things about the “colour of the grass”, “arrangement of the seats”, “staffs of the corresponding club” and “staff's duties for the corresponding club” and also the “grass quality". It's called concrete conclusion about the field. Now assume you are viewing HD satellite images of all fields on Earth. So you can observe all such fields and conclude about some common properties “field has green grasses” and it has “certain shape” and it has “a big seating stadium”. So, now we define a place P to be field if it has the following concluded properties. So, you widen the applicability of various logical conclusions about different fields and it is applicable to all such fields thus generalizing the concept of field. This is called abstraction of a particular notion. Associated set with such abstraction is called an abstract space. Pure mathematics deals with abstract spaces. Pure mathematicians do not care about application of their theory outside of mathematics actually.

Applied Mathematics :- It uses tools of pure mathematics which are abstract, to create tools which can be applied to solve real life problems. Generally, pure mathematics is at least 50 years ahead of applied mathematics. Applied mathematicians deal with concrete problems more. They don't care about theory much. They heavily relies in application of the theory rather building them. Their work is computation heavy.

So, that's my personal opinion entirely. Many people have different opinions obviously. I am open to all criticisms. Thanks ! Cheers !

1. Actuary. £100k to £250k
2. Chartered Accountants. ACA qualification. £75k to £350k. Partners in the big firms earn 7 figures.
3. Investment bankers. £100k start plus annual bonus.
4. Software Engineer. £30k to £250k
5. Maths teacher in schools. £30k to £65k. Head teachers earn 6-figures
6. Operational Research
7. Statistician in National Health Service
8. Offshore Engineering/ Riser Engineering
9. Station keeping on FPSO vessel
10. Certified Accountant, ACCA qualification. They get paid a less than ACA accountants generally speaking.
11. Tax advisor. CTA. Often combined with ACA qualification.
12. Gambling industry jobs
13. Meteorologist
14. Defence industry jobs ( if you can get Security clearance )

It depends on what your other skills are: certainly the degree alone will not be enough. You can do any of the following:

1. Actuary (this requires taking exams, and being really good with statistics)
2. Data Analyst
3. Software engineer (if you have enough programming experience)
4. Financial math
5. Government / Military / NSA

In general, having a degree in pure mathematics suggests that you are capable of learning independently, use logical reasoning, have creative problem solving skills, and can communicate deep ideas. This will make you stand out as a job candidate - provided that you have the minimum skills necessary for the job. This means you need to make sure you know programming, and statistics or financial math. The real world is not always interested in rigorous proof, but businesses have lots of hard problems that require someone who is used to deep thinking to solve the problem in ‘most cases’.

Of course, the other obvious choices are to go to graduate school, or become a high school teacher. In the end, it depends on what types of jobs you are willing to take. I know plenty of PhD mathematicians who went these directions, and they are generally happy with their decisions.

Applied mathematicians are translators.

Math is a language. Sometimes even though you can speak a language, you need someone fluent to help you fully understand what the Shakespeares of that language are really saying. Understanding pure math and turning it into practical applications can be like trying to understand how to build a pyramid from a stone engraving.

So to answer your question, Theoretical Engineering is the field where applied mathematicians have more options then pure mathematicians. We are able to take the theory the pure mathematicians are great at, and see a path towards real world use that the engineers crave.

The function of engineering is innovation and the rate of innovation over the next 50 years should be high, which will require more engineers, thus ensuring good compensation for engineering. However, many of the tasks performed by engineers over the last 50 years will be automated, thus mandating that the only successful future engineers will be those that have and use much more complex knowledge. The best engineering schools already require applicants to have finished mechanics with calculus and similar complexity and math in chemistry and/or biology, as well as a knowledge of coding. Their graduates will find that they will be lifetime students to keep up with a very rapidly evolving technical environment, but their ability to do so will assure a well compensated and well recognized career.

That depends on what you mean by ‘career associated with pure mathematics’.

Some of my friends from grad school refused to think about anything that was outside their ‘area of interest’. This is one extreme of the ‘pure mathematician’, and unless you are singularly good it will hurt your career, and there is very little chance of a job outside of academia unless the area happens to be something that a company or industry is particularly interested in. Even for people who stay in academia, this tunnel-vision approach to math generally isn’t good for their careers, and in my experience their utility to the department comes from their teaching and sitting on committees, as they have trouble attracting PhD students and getting grant money.

But I’m guessing you don’t mean that, since you included programmer as one of the careers. For the most part people who call themselves programmers don’t do much math - certainly in my case when I’m doing either math or programming I can’t do anything else.

For people who want to apply mathematics to real problems, there are lots of jobs. And there are even more jobs where people with mathematical training are good candidates (I know some people with PhD’s in math who went to work on Wall Street, and most of them said they no longer do math, but they find their work interesting and the certainly have made a LOT of money). But some places where you use math in your work, and likely will have the chance to do new math, are data science, government research, and finance, just to list some obvious categories.

In my experience jobs where you ‘do math’ are not that easy to get. The reason is that most people - even those who are good at math - aren’t very good at applying math, and having a mathematician sitting around being useless is demoralizing for everyone else. But the value of people who can apply math is very high in almost any technical job, so you can work your way into such a role if you go about it the right way (by being a team player, and making sure that the math problems you’re solving are valuable to your colleagues). You’ll need to do more than math in those roles, but there are no jobs even in academia where you only think about math (you have to at least communicate what you’ve figure out).

My dear friend even I completed civil engineering, its all based upon your interest , but the sad truth about civil engineering is placements , there are no proper placements when compared to other branches of engineering..

1. But the best thing is , civil engineer is an job creator ,rather than an job seeker.
2. Civil engineering is the oldest branch of engineering, it continued till now ..it will surely continue further..
3. If you inherit a true potential embedded with passion .., you can create miracles in the world's infrastructure.. so I hope you will have an great career as an civil engineer.. do well ….!

Your question makes a supposition that I think is entirely invalid. There is a huge amount of overlap between “pure science” and engineering in terms of the mental toolkit that is used and the language employed and it is entirely possible for people to move from engineering to science and vice-versa - indeed it happens all the time.

There are differences, of course and as a result there is a learning curve to making a transition between fields; this is true in any migration - it is probably less marked between e.g. Physics and Engineering than any other pair of fields.

To be fair, it is difficult for e.g. an experienced scientist to move into an equivalent engineering role, because they don’t have the years of experience of making products work, but the reverse is also true and the difference is a matter of education - someone who is bright and motivated can do it.

If you mean a paid career then essentially nothing. College professor is the only paycheck job. However, the paradox is that you will hardly have time to pursue the research you want, as the college will overwhelmingly demand teaching and other bureaucratic chores. Academia in America is a hamster wheel, plus you might get stuck for decades with people in your department whom you truly despise and every time you see him in the hall, the urge is strong to punch his teeth out. As an independent author, it will be hard to get published in a reputable journal as peer reviewed journals precisely don’t want independent authors, to protect the turf of college faculty. My adviser whose research field was not in the mainstream, maintained his paid position at the department AND established a working relationship with a lesser known publisher to publish his books, but he often had to type-set the pages himself, so essentially producing the pages publishing ready.

Apart from answering everything you can on Quora ? ;)

I’ll have a short answer based on what I do right now (actually, what I should do, but I’m about to leave office so don’t judge me).

I’m actually in a Technological Research Institute, and I’m working (it’s an internship ^^) on 5G technologies, especially new multi-antennas usage. And, oh boy, it has nothing to do with simulations of physical situations, or traditional engineering stuff. It’s not even applied maths, it’s just hard math everywhere. I struggle all day with analytic formulas for channels capacity, and I switch half time to pure algebra to harness the power of a given theorem about the (huge) matrix I juggle with.

I think it’s stated somewhere I can’t give all details about the fabulous maths I’m doing here, on the internet (patents, bla bla bla), but you have to know that when you start working around a formula like [math]\int_B\log_2(\det(I_{m_t}+\rho \bar{Q}\bar{H}^*\bar{H}))[/math] and consider matrixes variations, algebra considerations, to maximize an analytic expression, well… You. Have. Fun.

(Also cryptography is great ;) )

I am going to write the answer to your question segmented in two categories.

1) Careers with direct application of what you have studied:

• Immediately after your PhD you can go for post-doctoral positions at a University to further delve into the problem you have studied or look at its applications.

• Take up research position at a University or a research organization, private companies like Google, Microsoft Research, Govt. Intelligence Agencies and a ton other.

• Get into teaching as a Professor at Universities.

• If you can program, it opens Data Science for you and there is no end to what you can do or where you can be. It is the in thing right now.

2) Everything Else:

• Consulting: They like your thinking abilities and experience of dealing with abstraction and structures.

• Venture Capital: Same as above and the ability to understand algorithms behind some AI and machine learning startups.

• Equity research for Investment Banking firms

• Strategist for large companies after studying some Economics.

There is a lot you can do. Someone once told me, be good at what you do and there is no dearth of opportunities for you. There is a lot of opportunity for people with PhDs in any field to get into industry. And PhD in Mathematics is ranked up there. However, it is not a piece of cake to finish a PhD in pure mathematics. If you like mathematics, go for it. But it wouldn’t be ideal if career opportunities after the PhD will help you decide whether you should do it.

However, gone are the days where people joked about Math PhDs like this:

Q: What is the difference between a PhD in mathematics and a large pizza?

A: A large pizza can feed a family of four.